Linear representation theory of Janko group:J3

This article gives specific information, namely, linear representation theory, about a particular group, namely: Janko group:J3.
View linear representation theory of particular groups | View other specific information about Janko group:J3

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ and $\mathbb {C}$)	1, 85, 85, 323, 323, 324, 646, 646, 816, 1140, 1215, 1215, 1615, 1920, 1920, 1920, 1938, 1938, 2432, 2754, 3078 maximum: 3078, lcm: 50232960, number: 21, sum of squares: 50232960

GAP implementation

The degrees of irreducible representations can be computed using the CharacterDegrees and CharacterTable functions:

gap> CharacterDegrees(CharacterTable("J3"));
[ [ 1, 1 ], [ 85, 2 ], [ 323, 2 ], [ 324, 1 ], [ 646, 2 ], [ 816, 1 ], [ 1140, 1 ], [ 1215, 2 ], [ 1615, 1 ], 
[ 1920, 3 ], [ 1938, 2 ], [ 2432, 1 ],  [ 2754, 1 ], [ 3078, 1 ] ]

The full character table can be printed out using the Irr and CharacterTable functions: [SHOW MORE]

gap> Irr(CharacterTable("J3"));
[ Character( CharacterTable( "J3" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "J3" ),
    [ 85, 5, -5, 4, 1, 0, 0, -1, -1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, E(19)+E(19)^4+E(19)^5+E(19)^6+E(19)^7+E(19)^9+E(19)^11+E(19)^16+E(19)^17,
      E(19)^2+E(19)^3+E(19)^8+E(19)^10+E(19)^12+E(19)^13+E(19)^14+E(19)^15+E(19)^18 ] ), Character( CharacterTable( "J3" ),
    [ 85, 5, -5, 4, 1, 0, 0, -1, -1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, E(19)^2+E(19)^3+E(19)^8+E(19)^10+E(19)^12+E(19)^13+E(19)^14+E(19)^15+E(19)^18,
      E(19)+E(19)^4+E(19)^5+E(19)^6+E(19)^7+E(19)^9+E(19)^11+E(19)^16+E(19)^17 ] ), Character( CharacterTable( "J3" ), [ 323, 3, 8, -1, 3, -E(5)-E(5)^4,
      -E(5)^2-E(5)^3, 0, -1, -1, -1, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, 0, 0, 0 ] ), Character( CharacterTable( "J3" ),
    [ 323, 3, 8, -1, 3, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0, -1, -1, -1, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0, 0, 0, 0 ] ),
  Character( CharacterTable( "J3" ), [ 324, 4, 9, 0, 4, -1, -1, 1, 0, 0, 0, 0, -1, -1, 1, -1, -1, 1, 1, 1, 1 ] ), Character( CharacterTable( "J3" ),
    [ 646, -10, 7, -2, 2, -2*E(5)-2*E(5)^4, -2*E(5)^2-2*E(5)^3, -1, 0, 1, 1, 1, 0, 0, -1, E(5)+E(5)^4, E(5)^2+E(5)^3, 0, 0, 0, 0 ] ),
  Character( CharacterTable( "J3" ), [ 646, -10, 7, -2, 2, -2*E(5)^2-2*E(5)^3, -2*E(5)-2*E(5)^4, -1, 0, 1, 1, 1, 0, 0, -1, E(5)^2+E(5)^3, E(5)+E(5)^4, 0,
      0, 0, 0 ] ), Character( CharacterTable( "J3" ), [ 816, -16, 6, 6, 0, 1, 1, 2, 0, 0, 0, 0, -1, -1, 0, 1, 1, 0, 0, -1, -1 ] ),
  Character( CharacterTable( "J3" ), [ 1140, 20, 15, 6, -4, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 0 ] ), Character( CharacterTable( "J3" ),
    [ 1215, 15, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, E(17)+E(17)^2+E(17)^4+E(17)^8+E(17)^9+E(17)^13+E(17)^15+E(17)^16,
      E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14, -1, -1 ] ), Character( CharacterTable( "J3" ), [ 1215, 15, 0, 0, 3, 0, 0, 0, 1,
      0, 0, 0, 0, 0, 0, 0, 0, E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14, E(17)+E(17)^2+E(17)^4+E(17)^8+E(17)^9+E(17)^13+E(17)^15
         +E(17)^16, -1, -1 ] ), Character( CharacterTable( "J3" ), [ 1615, 15, -5, -5, -1, 0, 0, 3, -1, 1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( "J3" ), [ 1920, 0, 0, 3, 0, 0, 0, 0, 0, -E(9)^2+E(9)^4+E(9)^5-E(9)^7, -E(9)^2-2*E(9)^4-2*E(9)^5-E(9)^7,
      2*E(9)^2+E(9)^4+E(9)^5+2*E(9)^7, 0, 0, 0, 0, 0, -1, -1, 1, 1 ] ), Character( CharacterTable( "J3" ), [ 1920, 0, 0, 3, 0, 0, 0, 0, 0,
      2*E(9)^2+E(9)^4+E(9)^5+2*E(9)^7, -E(9)^2+E(9)^4+E(9)^5-E(9)^7, -E(9)^2-2*E(9)^4-2*E(9)^5-E(9)^7, 0, 0, 0, 0, 0, -1, -1, 1, 1 ] ),
  Character( CharacterTable( "J3" ), [ 1920, 0, 0, 3, 0, 0, 0, 0, 0, -E(9)^2-2*E(9)^4-2*E(9)^5-E(9)^7, 2*E(9)^2+E(9)^4+E(9)^5+2*E(9)^7,
      -E(9)^2+E(9)^4+E(9)^5-E(9)^7, 0, 0, 0, 0, 0, -1, -1, 1, 1 ] ), Character( CharacterTable( "J3" ), [ 1938, 2, 3, -6, -2, -E(5)-E(5)^4,
      -E(5)^2-E(5)^3, -1, 0, 0, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3, 1, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, 0, 0, 0 ] ), Character( CharacterTable( "J3" ),
    [ 1938, 2, 3, -6, -2, -E(5)^2-E(5)^3, -E(5)-E(5)^4, -1, 0, 0, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4, 1, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0, 0, 0, 0 ] ),
  Character( CharacterTable( "J3" ), [ 2432, 0, -16, 2, 0, 2, 2, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, 1, 1, 0, 0 ] ), Character( CharacterTable( "J3" ),
    [ 2754, -14, 9, 0, -2, -1, -1, 1, 0, 0, 0, 0, 1, 1, 1, -1, -1, 0, 0, -1, -1 ] ), Character( CharacterTable( "J3" ), [ 3078, -10, -9, 0, 2, -2, -2, -1,
      0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, 0, 0 ] ) ]

External links

• J3 on the Atlas of Finite Groups